4.2: Adding Integers
Introduction
Calculating Dives
On his first day in the Caribbean, Cameron completed his dive test and passed with flying colors. A dive test is done in a pool prior to diving. It lets the dive master know that you understand what you are doing and can handle yourself under the water. Scuba diving is exciting, but you have to know what you are doing to do it well.
The second day, Cameron and his Dad went for their first two dives. On the first dive, Cameron traveled to a depth of 25 feet. Then he and his Dad saw a stingray and followed it for a while and traveled down another 10 feet. Cameron took a few pictures of the stingray.
They traveled back to the boat for some surface time to eat, and rest, before going on the second dive. On his second dive, Cameron did a shallow dive of only 15 feet. He loved seeing the beautiful coral and even spotted a sea cucumber.
When they returned to the boat, Cameron began calculating his total depth and his total time for the day.
To calculate Cameron’s total depth for the day, you will need to know how to add integers. This lesson is all about this concept. It will give you all that you need to know to help Cameron calculate his total diving depths for the day.
What You Will Learn
In this lesson, you will learn to complete the following:
- Find sums of integers on a number line.
- Add two or more integers using absolute value.
- Evaluate variable expressions involving integer addition.
- Model and solve real-world problems using simple equations involving integer addition.
Teaching Time
I. Find Sums of Integers on a Number Line
In this lesson, 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.
In the last lesson, you saw examples where integers were represented on number lines. For example, 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 an example that shows how we can use a number line to model the addition of two positive integers.
Example
Use a number line to find the sum of \begin{align*}4+6\end{align*}.
You are adding two positive numbers. The positive numbers are to the right of zero on a number line.
So, to model \begin{align*}4+6\end{align*}, 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, \begin{align*}4+6=10\end{align*}.
Now, let's examine how to find the sum of two negative integers on a number line.
Example
Use a number line to find the sum of \begin{align*}-4+(-6)\end{align*}.
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 \begin{align*}-4+(-6)\end{align*}, 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, \begin{align*}-4+(-6)=-10\end{align*}.
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.
Example
Use a number line to find the sum of \begin{align*}4+(-6)\end{align*}.
Imagine starting at zero on the number line. You are adding a positive number, 4, to a negative number, -6.
To model \begin{align*}4+(-6)\end{align*}, 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 \begin{align*}4+(-6)=-2\end{align*}.
Example
Use a number line to find the sum of \begin{align*}-4+6\end{align*}.
Imagine starting at zero on the number line. You are adding a negative number, -4, to a positive number, 6.
So, to model \begin{align*}-4+6\end{align*}, 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, \begin{align*}-4+6=2\end{align*}.
Good! Noticing patterns will help as you add integers. Now let’s practice with a few examples.
4D. Lesson Exercises
Add by using a number line.
- \begin{align*}-5 + -8\end{align*}
- \begin{align*}-9 + 3\end{align*}
- \begin{align*}-9 + 12\end{align*}
Take a few minutes to check your work with a partner.
II. Add Two or More Integers Using Absolute Value
Using a number line is one strategy for adding integers now let’s look at another strategy.
Another strategy for adding integers involves using absolute values. An absolute value is the distance or the number of units that a number is from zero. Remember, with absolute value, the sign doesn’t matter. You will see the symbol \begin{align*}| \ |\end{align*} with an integer in the middle when absolute value is being represented.
Here are the steps to the absolute value strategy.
- To add two integers with the same sign, add their absolute values. Then give the answer the same sign as the two original integers.
- To add two integers with different signs, subtract the lesser absolute value from the greater absolute value. Then give the answer the same sign as the integer with the greater absolute value.
Take a few minutes to write these steps down in your notebook. Then continue.
How do we apply these steps?
Let’s look at an example and see what this looks like in action.
Example
Find the sum of \begin{align*}-13+(-12)\end{align*}.
Both integers being added have the same sign––a negative sign. So, add their absolute values.
Since \begin{align*}|-13|=13\end{align*} and \begin{align*}|-12|=12\end{align*}, add those values.
\begin{align*}13+12=25\end{align*}.
Give that answer, 25, the same sign as the original two integers––a negative sign.
So \begin{align*}-13+(-12)=-25\end{align*}.
Now you have seen this strategy with two negative numbers. Next, let’s see how it applies with a negative and a positive number.
Example
Find the sum of \begin{align*}13+(-12)\end{align*}.
The two integers being added have different signs. So, subtract their absolute values.
\begin{align*}|13|=13\end{align*} and \begin{align*}|-12|=12\end{align*}, subtract the lesser absolute value from the greater absolute value :
\begin{align*}13-12=1\end{align*}.
Give that answer, 1, the same sign as the integer with the greater absolute value. 13>12, so 13 has a greater absolute value than 12. Give the answer the sign of the number with the greater absolute value, in this case the 13, so the answer will have a positive sign.
So, \begin{align*}13+(-12)=1\end{align*}.
What happens if we are finding the sum of more than two integers?
You can use this same strategy to add three or more integers. When adding three or more integers, remember that the associative property of addition states that the grouping of numbers being added does not matter.
Example
Find the sum of \begin{align*}7+2+(-10)\end{align*}.
According to the associative property of addition, the integers being added can be grouped in any way. Here is one way to group numbers. Notice that we used brackets because parentheses are helpful when separating a negative sign and an addition sign. Brackets can mean the same thing as parentheses in these examples.
\begin{align*}[7+2]+(-10)\end{align*}
If you group the numbers this way, you will add \begin{align*}7+2\end{align*} first. Then you will add (-10) to that sum.
To add \begin{align*}7+2\end{align*}, first notice that both integers have the same sign––a positive sign. So, add their absolute values.
\begin{align*}|7|+|2|=7+2=9\end{align*}
Since the two original integers both had positive signs, give the sum a positive sign.
\begin{align*}[7+2]+(-10)=9+(-10)\end{align*}
Now, add \begin{align*}9+(-10)\end{align*}. Since both integers have different signs, find the absolute value of each integer.
\begin{align*}|9|=9\end{align*} and \begin{align*}|-10|=10\end{align*}, so subtract the lesser absolute value from the greater absolute value.
\begin{align*}10-9=1\end{align*}
Give that answer, 1, the same sign as the integer with the greater absolute value. \begin{align*}10>9\end{align*}, so -10 has a greater absolute value than 9. Give the answer a negative sign.
So, \begin{align*}7+2+(-10)=9+(-10)=-1\end{align*}.
Could I use a number line too?
Sure. A number line would have worked too. It just would have involved drawing a number line and then working through the math. Either way, you would still end up with the same answer.
Let’s practice using this method of adding integers.
4E. Lesson Exercises
- \begin{align*}-4 + 7 + -5\end{align*}
- \begin{align*}-9 + -12 + 8\end{align*}
- \begin{align*}-12 + 29 + -18\end{align*}
Check your answers with a friend. Is your work accurate? Correct any mistakes before moving on to the next section.
III. Evaluate Variable Expressions Involving Integer Addition
An expression is a number sentence that contains numbers and operations. An expression is not solved it is evaluated because there isn’t an equals sign in an expression. Some expressions include variables. These expressions are called variable expressions. A variable is a symbol or letter that is used to represent one or more numbers.
Variable expressions can be combined when there is a common term or a like term.
Example
\begin{align*}5x+7x\end{align*}
We can simplify or combine this variable expression because it has two common variables. The \begin{align*}x\end{align*}’s are common so we can find the sum of the variable expression.
The answer is \begin{align*}12x\end{align*}.
Example
\begin{align*}9y+8x\end{align*}
This variable expression can not be combined or simplified. It does not have like terms. The \begin{align*}x\end{align*} and the \begin{align*}y\end{align*} are different, so we can not do anything with this expression. It is in simplest form.
Sometimes, expressions will include both variables and integers. You can use what you know about how to add integers to help you find the value of expressions with variables.
Example
Find the sum \begin{align*}-4a+(-a)\end{align*}.
Since \begin{align*}-4a\end{align*} and \begin{align*}-a\end{align*} both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.
Both terms have the same sign, a negative sign. So, find the absolute values of both integers. Then add those absolute values to combine the terms. Remember, \begin{align*}-a=-1a\end{align*}.
\begin{align*}|-4|=4\end{align*} and \begin{align*}|-1|=1\end{align*}, so add \begin{align*}4a+1a=5a\end{align*}.
Since both terms had negative signs, give the answer a negative sign.
The sum of \begin{align*}-4a+(-a)\end{align*} is \begin{align*}-5a\end{align*}.
Example
Find the sum \begin{align*}-3t+9t\end{align*}.
Since \begin{align*}-3t\end{align*} and \begin{align*}9t\end{align*} both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.
Both like terms have different signs. So, find the absolute values of both integers. Subtract the term whose integer has the lesser absolute value from the term that has the greater absolute value.
\begin{align*}|-3|=3\end{align*} and \begin{align*}|9|=9\end{align*}, so subtract: \begin{align*}9t-3t=6t\end{align*}.
Since \begin{align*}9>3\end{align*}, and \begin{align*}9t\end{align*} has a positive sign, give the answer a positive sign.
The answer is \begin{align*}-3t+9t=6t\end{align*}.
Working with variable expressions may seem tricky, but if you first determine if you have like terms and then use the strategies you have learned for finding integer sums, you will be able to simplify each expression.
4F. Lesson Exercises
Simplify each variable expression.
- \begin{align*}-8x+ -5x\end{align*}
- \begin{align*}-19y+5y\end{align*}
- \begin{align*}-6y+2y+ -3y\end{align*}
Take a few minutes to check your work with a partner. Did you find a correct sum for the third problem?
IV. Model and Solve Real-World Problems Using Simple Equations Involving Integer Addition
Knowing how to add integers can also help you solve many problems in real life. To solve a real-world problem, write an expression or an equation that can be used to solve the problem, then solve.
Example 10
Molly lives in Alaska. The temperature outside her home at 6:00 A.M. one day last February was \begin{align*}-2^\circ F\end{align*}. Six hours later, the temperature had risen by \begin{align*}5^\circ F\end{align*}. What was the temperature six hours later?
The problem says that the temperature had risen six hours later. This means that the temperature had increased, so you should add. To find the new temperature, you can add the amount of the increase to the previous temperature.
You can find the temperature six hours later by using one of these equations.
\begin{align*}-2^\circ F + 5^\circ F & = ?\\ \text{or} \qquad -2+5&=?\end{align*}
Both integers have different signs. So, find the absolute values of both integers. Then subtract the integer with the lesser absolute value from the absolute value of the integer with the greater absolute value.
\begin{align*}|-2|=2\end{align*} and \begin{align*}|5|=5\end{align*}, so subtract: \begin{align*}5-2=3\end{align*}.
Since \begin{align*}5>2\end{align*}, and \begin{align*}5^\circ F\end{align*} has a positive sign, the temperature six hours later must be \begin{align*}3^\circ F\end{align*}.
Temperature is just one example of a real world problem involving integers. Think back to the original problem with our diver Cameron. We can figure out the sum of his depth now that we have learned to add integers.
Real-Life Example Completed
Calculating Dives
Here is the original problem once again. Reread it and underline any important information.
On his first day in the Caribbean, Cameron completed his dive test and passed with flying colors. A dive test is done in a pool prior to diving. It lets the dive master know that you understand what you are doing and can handle yourself under the water. Scuba diving is exciting, but you have to know what you are doing to do it well.
The second day, Cameron and his Dad went for their first two dives. On the first dive, Cameron traveled to a depth of 25 feet. Then he and his Dad saw a stingray and followed it for a while and traveled down another 10 feet. Cameron took a few pictures of the stingray.
They traveled back to the boat for some surface time to eat, and rest, before going on the second dive. On his second dive, Cameron did a shallow dive of only 15 feet. He loved seeing the beautiful coral and even spotted a sea cucumber.
When they returned to the boat, Cameron began calculating his total depth and his total time for the day.
To help Cameron calculate his total depth, we can write the following equation.
Depth of dive 1 + depth of dive 2 = total depth
Remember that depth is below the surface, so we use negative numbers to represent these integers.
On dive 1, Cameron went -25 ft and then -10 ft more
On dive 2, Cameron went -15 ft.
Now we can substitute these values into the equation.
\begin{align*}-25 + -10 + -15 = -50\end{align*}
Cameron’s total depth for the day was -50 feet.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Integer
- the set of whole numbers and their opposites.
- Sum
- the answer in an addition problem.
- Absolute Value
- the distance or number of units that an integer is from zero.
- Expression
- a number sentence that contains numbers, operations, and variables
- Variable
- a letter used to represent an unknown quantity.
- Variable Expression
- A phrase using numbers, operations and variables.
Technology Integration
Khan Academy Adding/Subtracting Negative Numbers
James Sousa, Adding Integers Using Absolute Values
James Sousa, Adding Integers Using a Number Line
James Sousa, Example of Adding Integers
James Sousa, Another Example of Adding Integers
Resources
You can learn more about junior scuba diving at www.padi.com.
Time to Practice
Directions: Add using a number line.
1. \begin{align*}2+5\end{align*}
2. \begin{align*}(-4)+4\end{align*}
3. \begin{align*}(-3)+(-3)\end{align*}
4. \begin{align*}8+(-6)\end{align*}
Directions: Use absolute values to find each sum.
5. \begin{align*}20+(-9)\end{align*}
6. \begin{align*}-30+(-40)\end{align*}
7. \begin{align*}-8+3+(-9)\end{align*}
8. \begin{align*}6+1+(-9)\end{align*}
9. \begin{align*}(-8)+-20\end{align*}
10. \begin{align*}(-6)+8+(-4)\end{align*}
11. \begin{align*}(2)+8+(-12)\end{align*}
12. \begin{align*}5+7+(-15)\end{align*}
Directions: Simplify each variable expression.
13. \begin{align*}7z+(-3z)\end{align*}
14. \begin{align*}(-10d)+(-d)+2\end{align*}
15. \begin{align*}8x+(-4x)-5\end{align*}
16. \begin{align*}7y+(-3y)\end{align*}
17. \begin{align*}16x+(-22x)\end{align*}
18. \begin{align*}5a+(-a)+7a\end{align*}
Directions: Solve each real-world problem.
19. A plane is flying at an altitude that is 2, 500 feet above sea level. If the plane increases its altitude by 500 feet, what will its new altitude be?
20. The temperature on a mountaintop, at midnight, was \begin{align*}-8^\circ F\end{align*}. By 3:00 A.M., the temperature had risen by \begin{align*}3^\circ F\end{align*}. What is the temperature at 3:00 A.M.?