4.5: Dividing Integers
Introduction
The Scuba Descent
Cameron and his new diving partner Gina are going to be buddies on a 40 foot dive. Gina is a new diver and is still learning to make a descent. Cameron can make a free descent quite easily. This means that he doesn’t hold onto anything as he descends to the appropriate depth. Gina will hold onto the anchor line as she descends. Then they will meet on the bottom.
Cameron has decided to go down with Gina. He will descend freely next to her, while she descends holding onto the rope. He looks at his watch and sets the timer before they descend.
When they reach the bottom, Cameron looks at his watch. He sees that the descent took them 2 minutes. Not bad at all considering that Gina is a beginner. Cameron and Gina meet up with the group and check in with the Dive Master. Then they are off for a beautiful dive!!
How far did Cameron and Gina descend per minute?
To answer this question, you will need to understand how to divide integers. Pay attention and you will be able to answer these questions at the end of the lesson.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following:
- Analyze patterns of quotients of integers with same and different signs, recognizing division by zero as undefined.
- Divide integers.
- Evaluate variable expressions involving integer division.
- Model and solve real-world problems using simple equations involving integer division.
Teaching Time
I. Analyze Patterns of Quotients of Integers with Same and Different Signs, Recognizing Division by Zero as Undefined
Another important step in learning how to compute with integers is learning how to divide them. You can look for patterns in a sequence of quotients just as you looked for patterns in a sequence of products in an earlier lesson. These patterns will help you to understand the rules for dividing integers.
Let’s look at some integer patterns with division. We are looking at quotients. A quotient is the answer in a division problem.
Example
Use a pattern to find the missing quotients below.
\begin{align*}6 \div 2 & = 3\\ 4 \div 2 & = 2\\ 2 \div 2 & = 1\\ 0 \div 2 & = 0\\ -2 \div 2 & = ?\\ -4 \div 2 & = ?\\\end{align*}
Look for a pattern among the quotients. Remember that a pattern has a rule that makes it repeat in a certain way. Look at the pattern below.
This pattern uses shapes and not numbers, but there is still a rule that applies, that makes the pattern repeat itself in the way that it does.
Now look at the number pattern.
You will see that you can subtract 1 from the previous quotient to find the next quotient. Remember, subtracting 1 is the same thing as adding its opposite, -1. Try adding -1 to the previous quotients to find the next quotients.
To find the quotient of \begin{align*}-2 \div 2\end{align*}, add \begin{align*}0+(-1)\end{align*}
\begin{align*}|0|=0\end{align*} and \begin{align*}|-1|=1\end{align*}, so subtract the lesser absolute value from the greater absolute value.
\begin{align*}1-0=1\end{align*}
The integer with the greater absolute value is -1, so give the answer a negative sign.
\begin{align*}0+(-1)=-1\end{align*}, so \begin{align*}-2 \div 2=-1\end{align*}
To find the quotient of \begin{align*}-4 \div 2\end{align*}, add \begin{align*}-1+(-1)\end{align*}
Both integers have the same sign, so add their absolute values.
\begin{align*}|-1|=1\end{align*}, so add
\begin{align*}1+1=2\end{align*}
Give that answer a negative sign.
\begin{align*}-1+(-1)=-2\end{align*}, so \begin{align*}-4 \div 2 =-2\end{align*}.
This shows the completed division facts.
\begin{align*}6 \div 2 & = 3\\ 4 \div 2 & = 2\\ 2 \div 2 & = 1\\ 0 \div 2 & = 0\\ -2 \div 2 & = -1\\ -4 \div 2 & = -2\\\end{align*}
Each quotient is still 1 less than the previous quotient.
What conclusions can we draw from this pattern?
You may notice the following.
- When a positive integer is divided by a positive integer, 2, the quotient is positive.
- When zero is divided by a positive integer, 2, the quotient is zero.
- When a negative integer is divided by a positive integer, 2, the quotient is negative.
These are the beginnings of our rules for dividing integers.
Let’s look at another pattern to complete these rules.
Example
Look at the number facts below. Analyze the pattern of quotients shown.
\begin{align*}&9 \div (-3) = -3\\ &6 \div (-2) = -3\\ &3 \div (-1) = -3\\ &0 \div 0 = undefined\\ &-3 \div (-1) = 3\\ &-6 \div (-2) = 3\\ &-9 \div (-3) = 3\end{align*}
What do you notice about these facts?
You may notice the following rules.
- When a positive integer is divided by a negative integer, the quotient is negative.
- When zero is divided by zero, the quotient is undefined, not zero. (Note: Any number divided by zero is considered undefined.)
- When a negative integer is divided by a negative integer, the quotient is positive.
Based on the patterns, here are the rules for dividing integers.
Take a few minutes to write these rules into your notebook. Notice that the rules for dividing integers are the same as the rules for multiplying integers.
II. Divide Integers
Now we can use these rules to divide integers. Just like with the rules for multiplying, becoming great at dividing integers will require that you memorize these rules.
Now, let’s apply these rules to dividing integers.
Example
Find the quotient \begin{align*}(-33) \div (-3)\end{align*}
To find this quotient, we need to divide two negative integers.
Divide the integers without paying attention to their signs. The quotient will be positive.
\begin{align*}(-33) \div (-3) = 33 \div 3 = 11\end{align*}
The quotient is 11.
Example
Find the quotient \begin{align*}(-20) \div 5\end{align*}.
To find this quotient, we need to divide two integers with different signs.
Divide the integers without paying attention to their signs. Give the quotient a negative sign because the signs are different, indicating a negative answer.
\begin{align*}20 \div 5 =4\end{align*}, so \begin{align*}(-20) \div 5 = -4\end{align*}.
The quotient is -4.
These problems used a division sign, but remember we can also show division using a fraction bar where the numerator is divided by the denominator.
Now, it’s time for you to practice applying these rules to figuring out quotients.
4L. Lesson Exercises
- \begin{align*}-12 \div -3\end{align*}
- \begin{align*}\frac{18}{-3}\end{align*}
- \begin{align*}-24 \div 8\end{align*}
Take a few minutes to check your work with a partner.
III. Evaluate Variable Expressions Involving Integer Division
You have used variable expressions with addition, subtraction and multiplication. Now we are going to apply division of integers with variable expressions. Remember that a variable expression is a math sentence that uses numbers, variables and operations.
Example
Find the value of this expression \begin{align*}-18x \div (-2)\end{align*}
It may help you to rewrite the problem like this using a fraction bar to divide. Now you can see which values can be divided
\begin{align*}\frac{-18x}{-2}\end{align*}
Then separate out the integers like this.
\begin{align*}\frac{-18x}{-2} = \frac{-18 \cdot x}{-2} = \frac{-18}{-2} \cdot x\end{align*}
Notice that we can divide the integers. The \begin{align*}x\end{align*} remains alone because there isn’t another \begin{align*}x\end{align*}. We separate it out. Then we divide the integer part and add the \begin{align*}x\end{align*} to the answer.
Since \begin{align*}-18 \div (-2) = 18 \div 2 = 9\end{align*} (remembering the rules to determine the sign of the answer), we know that \begin{align*}\frac{-18}{-2} \cdot x= 9 \cdot x = 9x\end{align*}
So, the value of the expression is \begin{align*}9x\end{align*}.
Let’s look at another example where there is a matching variable too.
Example
\begin{align*}-24y \div 2y\end{align*}
Next, we rewrite the expression using a fraction bar.
\begin{align*}\frac{-24y}{2y}\end{align*}
Now, we can separate the terms.
\begin{align*}&\frac{-24}{2} \cdot \frac{y}{y}\\ &-24 \div 2 = -12\end{align*}
\begin{align*}y \div y =1\end{align*} because the \begin{align*}y\end{align*}’s cancel each other out
\begin{align*}-12(1) = -12\end{align*}
The value of the expression is -12.
Example
\begin{align*}-18ab \div 9b\end{align*}
First, rewrite the expression using a fraction bar.
\begin{align*}\frac{-18ab}{-9b}\end{align*}
Next, separate out the terms.
\begin{align*}&\frac{-18}{-9} \cdot \frac{ab}{b}\\ &-18 \div -9 = 2\end{align*}
\begin{align*}ab \div b= a\end{align*} Notice that the \begin{align*}b\end{align*}’s cancel, but the a doesn’t. It is left as part of the final expression.
Our final answer is \begin{align*}2a\end{align*}.
4M. Lesson Exercises
- \begin{align*}-14a \div -7\end{align*}
- \begin{align*}28ab \div -7b\end{align*}
- \begin{align*}-6x \div -2\end{align*}
Take a few minutes to check your answers with a friend.
IV. Model and Solve Real-World Problems Using Simple Equations Involving Integer Division
We can apply these rules for dividing integers to real-world problems.
That’s a great question! To solve a real-world problem, write an expression or an equation that can be used to solve the problem. Then solve.
Example
On 3 consecutive plays, a football team lost a total of 30 yards. The team lost the same number of yards on each play. Represent the number of yards lost on each play as a negative integer.
First, represent the total number of yards lost as an integer.
Since the integer shows a loss of 30 yards, use a negative integer -30.
To represent the loss for each of the 3 plays, divide the integer representing the total number of yards lost by 3.
We write this equation and then fill in the given values
Total yards lost \begin{align*}\div\end{align*} number of plays = yds lost on each play
\begin{align*}-30 \div 3 =?\end{align*}
To find this quotient, we need to divide two integers with different signs.
Divide the integers without paying attention to their signs. Give the quotient a negative sign.
\begin{align*}30 \div 3 = 10\end{align*}, so \begin{align*}(-30) \div 3 = -10\end{align*}.
The integer -10 represents the number of yards lost on each play.
Now let’s apply what we have learned to the problem from the introduction.
Real-Life Example Completed
The Diving Descent
Here is the original problem once again. Reread it and underline any important information.
Cameron and his new diving partner Gina are going to be buddies on a 40 foot dive. Gina is a new diver and is still learning to make a descent. Cameron can make a free descent quite easily. This means that he doesn’t hold onto anything as he descends to the appropriate depth. Gina will hold onto the anchor line as she descends. Then they will meet on the bottom.
Cameron has decided to go down with Gina. He will descend freely next to Gina, while she descends holding onto the rope. He looks at his watch and sets the timer before they descend.
When they reach the bottom, Cameron looks at his watch. He sees that the descent took them 2 minutes. Not bad at all considering that Gina is a beginner. Cameron and Gina meet up with the group and check in with the Dive Master. Then they are off for a beautiful dive!!
How far did Cameron and Gina descend per minute?
First, we need to write integers to represent the depth and the time.
-40 feet is the depth
2 minutes is the time
We divide the depth by the time to find out the number of feet per minute.
\begin{align*}-40 \div 2 = -20\end{align*}
Cameron and Gina descended -20 feet per minute.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Quotient
- the answer in a division problem.
- Undefined
- when an integer is divided by 0, the answer is undefined.
- Variable Expression
- a math sentence with numbers, operations and variables
Technology Integration
Khan Academy Multiplying and Dividing Negative Numbers
James Sousa, Division of Integers: The Basics
James Sousa, Example of Dividing Integers
James Sousa, Multiplying and Dividing Involving Zero
Time to Practice
Directions: Analyze the patterns and find the missing quotients.
1. \begin{align*}24 \div 8 = 3\!\\ 16 \div 8 = 2\!\\ 8 \div 8 = 1\!\\ 0 \div 8 = 0\!\\ -8 \div 8 = ?\!\\ -16 \div 8 = ?\end{align*}
2. \begin{align*}21 \div (-3) = -7\!\\ 14 \div (-2) = -7\!\\ 7 \div (-1) = ?\!\\ 0 \div 0 = ?\!\\ -7 \div (-1) = ?\!\\ -14 \div (-2) = 7\!\\ -21 \div (-3) = 7\end{align*}
Directions: Find each quotient.
3. \begin{align*}48 \div 8\end{align*}
4. \begin{align*}64 \div (-8)\end{align*}
5. \begin{align*}-28 \div (-4)\end{align*}
6. \begin{align*}-35 \div 7\end{align*}
7. \begin{align*}-80 \div (-4)\end{align*}
8. \begin{align*}-50 \div 10\end{align*}
9. \begin{align*}-18 \div -2\end{align*}
10. \begin{align*}42 \div -6\end{align*}
11. \begin{align*}-72 \div 9\end{align*}
12. \begin{align*}-48 \div -12\end{align*}
13. \begin{align*}-16 \div 4\end{align*}
14. \begin{align*}-22 \div -11\end{align*}
15. \begin{align*}72 \div -12\end{align*}
Directions: Find each quotient with variable expressions.
16. \begin{align*}36t \div (-9)\end{align*}
17. \begin{align*}-56n \div (-7)\end{align*}
18. \begin{align*}-22n \div -11n\end{align*}
19. \begin{align*}-28n \div 7\end{align*}
20. \begin{align*}18xy \div 2x\end{align*}
Directions: Solve the following real-world example.
21. Company X lost a total of $1,200 during its first 4 months in business. Suppose the company lost the same number of dollars each month. Represent the number of dollars lost each month as a negative integer.
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