11.7: Sums of Integers with the Same Sign
Have you ever sewed something and had to take out a bunch of stitches?
After finishing her letter to her pen pal, Callie began working on sewing her sweater. She had caught it on a hook at school and tore a small hole in the right sleeve. Callie sewed fourteen stitches and then added another ten to that. She thought that she had picked a thread that would match the sweater exactly. When she looked at her work, she discovered that she had not matched the sweater with her choice of thread.
Callie took out 10 stitches, and then she took out another 14 stitches.
How many did she take out altogether?
Can you write this problem using negative numbers?
This Concept will show you just how to do that.
Guidance
When you add integers with the same sign, you think in terms of gains OR in terms of losses, but not both at once. You don’t have losses and gains just one or the other.
5 + 9 = ____
These two integers are both positive. We can add them just as we would any whole numbers. Five plus nine is fourteen. That is our answer.
5 + 9 = 14
The answer is 14.
A positive plus a positive is still positive. The sign stays the same when you add two integers that have the same sign.
What about a negative integer plus a negative integer?
A negative integer is a loss. A loss plus another loss is more loss. As before, the sign stays the same when you add two integers that have the same sign.
-9 + -8 = ____
A negative nine plus a negative eight is a negative seventeen.
-9 + -8 = -17
The answer is -17.
We can notice two things.
- The sign stays the same when you add two integers that have the same sign.
- You can add the integers just as you would whole numbers and keep the sign the same.
A gain and a gain is more gain.
A loss and a loss is more loss.
Practice adding integers with the same sign.
Example A
-9 + -11 = ____
Solution: -20
Example B
-8 + -13 = ____
Solution: -21
Example C
17 + 12 = ____
Solution: 29
Now back to Callie and the sweater.
After finishing her letter to her pen pal, Callie began working on sewing her sweater. She had caught it on a hook at school and tore a small hole in the right sleeve. Callie sewed fourteen stitches and then added another ten to that. She thought that she had picked a thread that would match the sweater exactly. When she looked at her work, she discovered that she had not matched the sweater with her choice of thread.
Callie took out 10 stitches, and then she took out another 14 stitches.
How many did she take out altogether?
Can you write this problem using negative numbers?
The first part of this problem is simple, we can add 10 and 14.
\begin{align*}10 + 14 = 24\end{align*}
To write this problem using negative numbers, we add a negative sign to each value. This shows that Callie "took out" the stitches and did not add them.
\begin{align*}-10 + -14 = -24\end{align*}
This is our answer.
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.
- Absolute Value
- the number of units that an integer is from zero. The sign does not make a difference.
Guided Practice
Here is one for you to try on your own.
Add the following integers.
\begin{align*}-7 + -12 + -36\end{align*}
Answer
Since all of these values are negative, we can simply add the values and then make our answer negative.
\begin{align*}7 + 12 + 36 = 55\end{align*}
The answer is \begin{align*}-55\end{align*}
Video Review
Here is a video for review.
James Sousa, Example of Adding Integers
Practice
Directions: Add the following integers.
1. \begin{align*}6 + 9\end{align*}
2. \begin{align*}-6 + -9\end{align*}
3. \begin{align*}-16 + -9\end{align*}
4. \begin{align*}16 + 19\end{align*}
5. \begin{align*}23 + 90\end{align*}
6. \begin{align*}86 + 99\end{align*}
7. \begin{align*}-96 + -9\end{align*}
8. \begin{align*}-76 + -29\end{align*}
9. \begin{align*}-336 + -9\end{align*}
10. \begin{align*}206 + 89\end{align*}
11. \begin{align*}-6 + -9 + -17\end{align*}
12. \begin{align*}-34 + -9 + -12\end{align*}
13. \begin{align*}-96 + -19+ -8\end{align*}
14. \begin{align*}-76 + -29 + -11\end{align*}
15. \begin{align*}-36 + -89 + -33\end{align*}
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Absolute Value
The absolute value of a number is the distance the number is from zero. Absolute values are never negative.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...Image Attributions
Here you'll learn to add integers with the same sign.