4.1: Comparing and Ordering Integers
Introduction
Dive Depths
Cameron and his parents are all scuba divers. Cameron learned to scuba dive two years ago when he was eleven. Kids between the ages of 11 and 14 can become certified junior divers through an organization called PADI. Since Cameron learned to dive, he has looked forward to his family’s diving vacation each year when they all take off to someplace warm to scuba dive.
One week before this year's big trip to the Caribbean, Cameron began looking through his dive book. A dive book is a book where divers keep track of their dives. They chart the depth that they went, the time they were underwater, and anything cool that they saw.
As a junior diver, Cameron is only allowed to travel to a maximum depth of 40 feet.
Here are Cameron’s dive depths from his last trip when he went diving in Jamaica.
15 feet deep
40 feet deep
25 feet deep
36 feet deep
30 feet deep
Integers can help a scuba divers in a real-world situation like this. Since Cameron traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.
To do this, you will need to know about integers. Pay attention to this lesson, and at the end of the lesson you will know how to help Cameron write out his diving depths from least to greatest.
What You Will Learn
In this lesson, you will learn how to complete the following:
- Write integers representing situations of increase/decrease, profit/loss, above/below, etc.
- Identify absolute value and opposites of given integers, recognizing zero as neither positive nor negative.
- Compare and order integers on a number line.
- Compare and order integers using inequality symbols.
Teaching Time
I. Write Integers Representing Situations of Increase/Decrease, Profit/Loss, Above/Below
Numbers can be classified in many different ways. For example, we can classify numbers as whole numbers, fractions, and decimals.
Some numbers can be classified as integers. Integers include the positive whole numbers (1, 2, 3, 4, 5, ...), their opposites (-1, -2, -3, -4, -5, ...) and zero.
This number line shows the integers from -5 to 5.
Look at the number line. The negative integers (-1, -2, -3, -4, and, -5) are to the left of 0, so their values are less than 0. The positive integers (1, 2, 3, 4, and, 5) are to the right of 0, so their values are greater than 0.
We can use numbers to describe real world situations and using integers can assist us with this as well. Let's take a look at how integers can help us describe real-world situations.
We can use integers to represent many real-world situations, such as:
- Increases and decreases in temperature.
- Profit and loss of money
- Locations above and below sea level.
First, let's take a look at how integers can help us represent temperatures.
Example
The temperature outside a ski lodge was \begin{align*}3^\circ F\end{align*} below \begin{align*}0^\circ F\end{align*}. Express that temperature with an integer.
To write this as an integer, we can think of a thermometer. A thermometer is just a vertical number line. Find the mark for \begin{align*}0^\circ F\end{align*} on this thermometer. With your finger, count \begin{align*}3^\circ F\end{align*} below that mark.
Your finger will point to \begin{align*}-3^\circ F\end{align*}. That is how the temperature \begin{align*}3^\circ F\end{align*} below \begin{align*}0^\circ F\end{align*} can be expressed as an integer. Think of the winter and the summer. When it is very cold or very hot, integers help us to understand how cold or how hot it is.
Example
A fisherman is sitting 2 feet above the surface of a lake on a boat. The hook on his fishing pole is floating 6 feet below the lake's surface. Use integers to represent the position of the fisherman and his hook.
Think of a vertical number line.
The surface of the lake can be represented by the integer, 0.
The fisherman is sitting 2 feet above the surface. You can represent this as +2 or 2.
The hook is floating 6 feet below the surface. You can represent this as -6.
Wow! Working with a picture certainly helps to make it very clear!!
Example
Mr. Marsh invested in the stock market and had a loss of $45 yesterday. Mrs. Marsh also invested in the stock market. Her investment showed a gain of $20 yesterday. Represent these situations with integers.
Think of a number line from -$50 to $50. The $0 mark represents neither a gain nor a loss on an investment.
Use a negative integer to represent a loss. Mr. Marsh lost $45 on his investment. This can be represented as -$45.
Use a positive integer to represent a gain. Mrs. Marsh's investment showed a gain of $20. This can be represented as +$20 or $20, because positive integers can be written with or without a positive (+) sign.
Do you get the idea? You can look for key words that indicate a positive or a negative number. When you look at a problem, identify any words that might tell you whether you are going to write a positive or a negative number. Think back at the last three examples and write down any key words that you notice.
4A. Lesson Exercises
Write an integer for each example.
- An increase of $200.00
- Down 10%
- 50 feet below sea level
Check your answers with a partner.
II. Identify Absolute Value and Opposites of Given Integers, Recognizing Zero as Neither Positive nor Negative
Sometimes, when we look at an integer, we aren’t concerned with whether it is positive or negative, but we are interested in how far that number is from zero. Think about water. You might not be concerned about whether the depth of a treasure chest is positive or negative simply how far it is from the surface.
This is where absolute value comes in.
What is absolute value?
The absolute value of a number is its distance from zero on the number line.
We use symbols to represent the absolute value of a number. For example, we write the absolute value of 3 as \begin{align*}|3|\end{align*}.
Writing an absolute value is very simple you just leave off the positive or negative sign and simply count the number of units that an integer is from zero.
Example
Find the absolute value of 3. Then determine what other integer has an absolute value equal to \begin{align*}|3|\end{align*}.
Look at the positive integer, 3, on the number line. It is 3 units from zero on the number line, so it has an absolute value of 3.
Now that you have found the absolute value of 3, we can find another integer with the same absolute value. Remember that with absolute value you are concerned with the distance an integer is from zero and not with the sign.
Here is how we find another integer that is exactly 3 units from 0 on the number line. The negative integer, -3, is also 3 units from zero on the number line, so it has an absolute value of 3 also.
So, \begin{align*}|3|=|-3|=3\end{align*}.
This example shows that the positive integer, 3, and its opposite, -3, have the same absolute value. On a number line, opposites are found on opposite sides of zero. They are each the same distance from zero on the number line. Because of this, any integer and its opposite will always have the same absolute value. To find the opposite of an integer, change the sign of the integer.
Just like we can find the absolute value of a number, we can also find the opposite of a number.
Example
Find the opposite of each of these numbers: -16 and 900.
-16 is a negative integer. We can change the negative sign to a positive sign to find its opposite. The opposite of -16 is +16 or 16.
900 is the same thing as +900. We can change the positive sign to a negative sign to find its opposite. So, the opposite of 900 is -900.
4B. Lesson Exercises
Find the absolute value of each number.
- \begin{align*}|22|\end{align*}
- \begin{align*}|-222|\end{align*}
- Find the opposite of -18.
Take a few minutes to check your answers with a neighbor.
III. Compare and Order Integers on a Number Line
Now that you know about absolute value, opposites, zero and integers, we can work on learning how to compare and order integers. Often, the easiest way to do this is by using a number line.
Remember, a number line shows numbers ordered from least to greatest. So, on a number line, the further a number is to the right, the greater its value.
This can be a little tricky because when you look at a number line -222, you might think that it is larger number, but it isn’t.
How can this be?
Consider an example. If you were thinking about how much money you had, a positive amount could be a bank account balance. However, if spent too much money, you would owe more than you had. This could be considered a negative number. Therefore, if your bank account balance was -$222 this means have even less money than if your bank account balance was -$22.The best way to think about it is the farther that a negative number is on the left side of the number line, the smaller that number is.
Example
Order these numbers on a number line. Then determine which one is the greatest and which is the least: -12, 5, -6, -9, 10, 7
First, let’s draw a number line and plot the numbers on it.
Now look at the values on this number line and where they are located.
You can see that the number furthest to the right is 10. That is the largest number. The number furthest to the left is -12. That is the smallest number.
Example
Order these integers from least to greatest: -6, 0, 5, -1.
To help you order these integers, draw a number line from -6 to 6. Then plot points for -6, 0, 5 and -1.
The numbers, ordered from least to greatest, are -6, -1, 0, 5.
Once we have an idea about which numbers are smaller or larger, we can work on comparing them.
4C. Lesson Exercises
Use a number line and write these numbers in order from least to greatest.
- -4, 2, 8, 9, -11, -5
- 6, -16, 7, -22, 1, 4
- -3, -2, -7, -12, -1
Take a few minutes to check your work with a partner. Is your work accurate?
IV. Compare and Order Integers Using Inequality Symbols
Once we understand how to determine which integers are greater, we can compare the integers using symbols. We can use these inequality symbols to compare and order integers.
> means is greater than.
< means is less than.
= means is equal to.
\begin{align*}\neq\end{align*} means is not equal to.
Example
Choose the inequality symbol that goes in the blank to make each statement true.
a. \begin{align*}-2 \ \underline{\;\;\;\;\;\;} \ -4\end{align*}
b. \begin{align*}-4 \ \underline{\;\;\;\;\;\;} \ 4\end{align*}
c. \begin{align*}|-4| \ \underline{\;\;\;\;\;\;} \ 4\end{align*}
To help you compare these values, draw a number line from -5 to 5, like this.
Consider statement a.
-2 is to the right of -4 on the number line. So, -2 is greater than -4.
The symbol > goes in the blank, because \begin{align*}-2 > -4\end{align*}.
Consider statement b.
-4 is to the left of 4 on the number line. So, -4 is less than 4.
The symbol < goes in the blank, because \begin{align*}-4 < 4\end{align*}.
Consider statement c.
\begin{align*}|-4|\end{align*} is 4, because -4 is 4 units from zero on the number line.
Since \begin{align*}|-4| = 4\end{align*}, the symbol = goes in the blank.
Now we can return to the original problem and work on helping Cameron with his diving dilemma.
Real-Life Example Completed
Diving Depths
Here is the original problem once again. Reread it and underline any important information.
Cameron and his parents are all scuba divers. Cameron learned to scuba dive two years ago when he was eleven. Kids between the ages of 11 and 14 can become certified junior divers through an organization called PADI. Since Cameron learned to dive, he has looked forward to his family’s diving vacation each year when they all take off to someplace warm to scuba dive.
One week before this year's big trip to the Caribbean, Cameron began looking through his dive book. A dive book is a book where divers keep track of their dives. They chart the depth that they went, the time they were underwater, and anything cool that they saw.
As a junior diver, Cameron is only allowed to travel to a maximum depth of 40 feet.
Here are Cameron’s dive depths from his last trip when he went diving in Jamaica.
15 feet deep
40 feet deep
25 feet deep
36 feet deep
30 feet deep
Integers can help scuba divers in a real-world situation like this. Since Cameron traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.
First, let’s write each of Cameron’s depths as an integer. Depth is a word that tells us that we are going below the surface of the water. If the surface is zero, then anything below the surface would be represented by a negative number. Cameron’s depths are all negative numbers.
-25
-30
-15
-36
-40
We can order these integers from least to greatest by thinking of the deepest dive as the least and the dive closest to the surface, zero, as the greatest.
-40, -36, -30, -25, -15
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Whole Numbers
- the positive counting numbers including 0.
- Fractions
- parts of a whole written with a numerator and denominator.
- Decimal
- parts of a whole written with a decimal point using place value.
- Integers
- positive whole numbers and their opposites. Positive and negative numbers
- Opposites
- Negative numbers have a positive partner. Positive numbers have a negative partner.
- Absolute Value
- the distance that a number is from zero.
Technology Integration
Khan Academy Negative Numbers Introduction
Comparing and Ordering Integers
James Sousa, Introduction to Integers
James Sousa, Example of Determining the Opposite of Integers
James Sousa, Example of Simplifying the Opposites of Integers
James Sousa, Comparing Integers Using Inequalities
James Sousa, Example of Ordering Integers from Least to Greatest
James Sousa, Example of Integer Application: Feet Below Sea Level
Resources
You can learn more about becoming a junior scuba diver at www.padi.com.
Time to Practice
Directions: Write each as an integer.
1. 10 degrees below zero
2. \begin{align*}50^\circ F\end{align*}
3. A loss of $20.00
4. 35 feet below the surface
5. 120 feet below sea level
6. An altitude of 15,000 feet
Directions: Write the opposite of each integer.
7. 20
8. -7
9. 22
10. -34
11. 0
12. -9
13. 14
14. 25
Directions: Find the absolute value of each number.
15. \begin{align*}|13|\end{align*}
16. \begin{align*}|-11|\end{align*}
17. \begin{align*}|-5|\end{align*}
18. \begin{align*}|17|\end{align*}
19. \begin{align*}|-9|\end{align*}
Directions: Compare the following integers using inequality symbols.
20. \begin{align*}-17 \ \underline{\;\;\;\;\;\;} \ -19\end{align*}
21. \begin{align*}-9 \ \underline{\;\;\;\;\;\;} \ -11\end{align*}
22. \begin{align*}4 \ \underline{\;\;\;\;\;\;} \ -3\end{align*}
23. \begin{align*}5 \ \underline{\;\;\;\;\;\;} \ 7\end{align*}
24. \begin{align*}9 \ \underline{\;\;\;\;\;\;} \ -9\end{align*}
25. \begin{align*}-12 \ \underline{\;\;\;\;\;\;} \ -23\end{align*}
26. \begin{align*}|-9| \ \underline{\;\;\;\;\;\;} \ |8|\end{align*}
27. \begin{align*}|9| \ \underline{\;\;\;\;\;\;} \ |-9|\end{align*}
28. \begin{align*}|-2| \ \underline{\;\;\;\;\;\;} \ |-7|\end{align*}