11.1: Comparing Integers
Introduction
The Pen pal Project
The students in Mrs. Harris’ Language Arts class in New York City are participating in a pen pal project. Last summer, Mrs. Harris went to Auckland, New Zealand. While there, she participated in a conference with other educators and became good friends with another teacher from New Zealand. Mrs. Harris has arranged for her students to exchange letters with the other students in New Zealand. They will be able to do some of their correspondence on email, and some will be done the old fashioned way. The students are very excited. Once a week, each class will be given some topic to investigate so that they can learn about the similarities and differences between life in New Zealand and life in the United States.
The first thing that students are being asked to look at is temperature. New Zealand and New York are in different parts of the world. The students need to figure out the average high and low winter temperatures in New Zealand and compare it with the average high and low winter temperatures in New York.
The first thing that the students notice is that the winter is New Zealand is during the months of June, July and August - the opposite of New York. The students find this very funny and Mrs. Harris takes the opportunity to teach the students about different hemispheres.
The average winter temperature in New York is:
High \begin{align*}= 40^\circ \ F\end{align*}
Low \begin{align*}= 28^\circ \ F\end{align*}
The average winter temperature is New Zealand is:
High \begin{align*}= 59^\circ \ F\end{align*}
Low \begin{align*}= 48^\circ \ F\end{align*}
One of the students, Karen, does this calculation.
The difference between the high and low winter temperatures in New York and the high and low winter temperatures in New Zealand is
\begin{align*}40 - 59 &= -19^\circ\\ 28 - 48 &= -20^\circ\end{align*}
Joey looked at her calculation and was puzzled.
“What does that mean?” he asked.
“Those are negative numbers,” Karen explained.
“Negative what?”
“Negative numbers,” Karen began.
Let’s stop there. This chapter is all about integers. When learning about integers, you will learn about positive and negative numbers. Pay attention to this lesson and Karen will finish explaining at the end of the lesson.
What You Will Learn
By the end of this lesson you will be able to properly the following skills:
- Write integers representing situations of increase/decrease, profit/loss, above/below, etc.
- Identify opposites of given integers
- Compare and order integers on a number line.
- Compare and order positive and negative fractions and decimals.
Teaching Time
I. Write Integers Representing Situations of Increase/Decrease, Profit/Loss, Above/Below, etc.
In mathematics so far, you have learned about some different kinds of numbers. You learned about whole numbers, fractions, decimals and percentages. In this lesson, you are going to learn about integers.
What is an integer?
An integer is a member of the set of whole numbers and their opposites.
To better understand integers, let’s first think about some real life situations where you might have seen integers used before.
Integers can be thought of as positive and negative numbers.
Let’s use temperature as an example.
If you look at this thermometer, you will see that the temperature is \begin{align*}79^\circ\end{align*}. This is 79 degrees above zero.
When we think about temperature, we use integers all the time. You might have heard someone talk about temperatures above or below zero. A temperature that is above zero is positive. A temperature that is below zero is negative.
Integers are positive and negative whole numbers. Said another way, they are whole numbers and their opposites.
How can we write integers?
When writing an integer, we can use a + sign or a - sign. A + sign can be used for a number above zero or a positive number. A positive number can also be written without the + sign. A negative number should be written using the - sign.
In the first example, we looked at integers having to do with temperature.
Example
5 degrees below zero
The words “below zero” let you know that this integer is a negative number. Since it is five degrees below zero, we can write the integer negative five.
The answer is -5.
Example
83 degrees
This temperature does not have a "-" sign in front of it, and does not say "below zero" either, it is a positive number. We write the integer positive eighty-three.
The answer is 83 or +83.
There are other real-life situations that use positive and negative numbers too.
One example is with money. A loss of money would be a negative integer. A gain of money is a positive integer.
Example
Jeff earned two hundred dollars working at the farm stand.
Jeff’s money is a gain. He “earned” it. That means that he had an increase in money and not a decrease. An increase or a gain is a positive integer. We can write the two hundred dollars as a positive integer.
The answer is $200.00 or +$200.00.
Example
Sasha spent $45.00 at the clothing store.
Since Sasha spent this money, it is a loss. Therefore, we can say that Sasha’s $45.00 is a negative integer. A negative integer can show a loss of money.
The answer is -$45.00.
The stock market is another real-life situation that uses positive and negative integers all the time.
In this picture the red arrow represents a gain and would be written as a positive integer. The purple arrow represents a loss and would be written as a negative integer.
What are some other key words that mean positive or negative integers?
We have already talked about losses and gains and above zero and below zero.
Profit and loss are two other words that mean positive or negative integers. Profit means positive and loss means negative.
An earning is a positive number.
Spending is a negative number.
Practice writing integers by using key words. Write an integer for each example.
- 50 feet below sea level
- $100.00 was spent
- A heat wave of \begin{align*}98^\circ\end{align*}
Take a few minutes to check your work with a peer.
II. Identify Opposites of Given Integers
An integer is the set of whole numbers and its opposites. This means that every integer has an opposite integer.
Here is what the set looks like.
....-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...
You see that each integer has an opposite value and the three dots at each end means that the set of whole numbers continues indefinitely in both a positive and negative direction.
How do we write opposites of given integers?
Well, it is actually very simple. Change the sign!
Let’s look at an example.
Example
Write the opposite of -15.
The answer would be +15 or 15.
Example
Write the opposite of 2200.
The answer would be -2200.
Practice writing the opposite integer for each example listed below.
- A loss of 34.
- 998
- 67 feet above sea level
Take a few minutes to check your work with a peer.
III. Compare and Order Integers on a Number Line
Now that you know about positive and negative integers, you can learn about comparing them. When we compare and order positive and negative integers, we use greater than, less than or equal to, or we write them in order from least to greatest or from greatest to least. Let’s start by learning about comparing integers.
How do we compare integers?
There are a couple of important things to consider when comparing integers.
1. A positive number is ALWAYS greater than a negative number.
The more positive a number, the greater it is. Let’s look at an example.
Example
-6 ____ 2
Negative six is below zero. Two is above zero. Two is greater than negative six.
-6 < 2
2. If two numbers are positive, the larger number is greater.
Example
17____10
Seventeen is greater than 10.
17 > 10
3. If two numbers are negative, the number closer to zero is greater than the other.
For two negative numbers, you have to think backwards. The larger the number the greater the loss is. The greater the loss, the smaller the number. Think about the number and its relationship to zero. This will help you determine whether it is greater than or less than.
Example
-25 ____ -36
Negative 25 is closer to zero than -36. It is the greater number.
-25 > -36
It does seem a bit backwards, but that is the way it is. You can think of it another way by looking at a number line.
Example
-9 ____ -3
Let’s look at a number line to compare these values.
If you look at where negative nine is compared with negative three, you can see that negative three is closer to zero. Negative three is greater than negative nine.
-9 < -3
Use greater than, less than or equal to and compare each example.
- -6 ____ 8
- -99 ____ -9
- 12 ____ 6
Take a few minutes to check your answers with a friend.
Now that you know how to compare two integers, we can work on ordering them from least to greatest and from greatest to least.
First, let’s look at a number line. This is a useful tool for ordering integers.
You can see that -10 is the lowest number on the number line and 10 is the greatest number on the number line.
The other numbers fall in between these two integers.
Let’s practice ordering by using a number line. Here is an example.
Example
Write from least to greatest.
-3, 2, -9, 5, 6, 0
We start with the integer that is the least. This is the number that is the MOST negative, which would be the number furthest to the left on the number line above. Then we work our way up the highest positive number, which is the number furthest to the right.
-9, -3, 0, 2, 5, 6
Using a number line can help us picture the value of each integer. Then we write them in order from least to greatest or from greatest to least according to the directions.
Practice writing the following sets of integers in order from least to greatest.
- 7, 4, 2, -19, 0, -12, 11
- 4, -4, 5, 7, 0, 10, -7
Take a few minutes to compare your answer with a friend’s work. Is your work accurate?
IV. Compare and Order Positive and Negative Fractions and Decimals
We have been working with the set of integers. Integers are positive and negative whole numbers. However, we can also have positive and negative fractions and decimals. These positive and negative fractions and decimals are not members of the set of integers. They are rational numbers and you will work more with them next year.
That being said, we can still order and compare positive and negative decimals and fractions.
How do we do this?
Let’s look at an example and work through comparing and ordering positive and negative decimals and fractions.
Example
\begin{align*} \frac{-1}{2} \underline{\;\;\;\;\;\;\;\;} \frac{-3}{4}\end{align*}
If we want to compare negative one-half and negative three-fourths, we have to think of which fraction is closer to zero. Negative one-half is smaller than negative three-fourths. Remember when we work with negative numbers that the smaller negative number is greater.
\begin{align*}\frac{-1}{2} > \frac{-3}{4}\end{align*}
We can use a number line to help us here too.
Here is another way to look at these numbers using a number line. You can see here that \begin{align*}\frac{-1}{2}\end{align*} is closer to 0 than \begin{align*}\frac{-3}{4}\end{align*} so it is the LESS negative and that means it is a greater value. This kind of thinking will work with any negative fractions. Remember that a positive fraction is ALWAYS greater than a negative fraction.
What about negative and positive decimals?
Negative and positive decimals can be compared just like fractions. Decimals are a part of a whole just like fractions are a part of a whole. Therefore, a positive decimal is ALWAYS greater than a negative decimal. When you have two negative decimals, the one closer to zero is always greater. The farther a negative decimal is from zero, the smaller its value.
Example
.45 ____ -.18
A positive number is always greater than a negative number. This also holds true for decimals and fractions.
.45 > -.18
Example
-.29 ____ -.56
The smaller the number the closer it is to zero.
-.29 > -.56
Now it’s time for you to practice. Compare the following negative and positive decimals.
- -.98 ____ -.88
- \begin{align*}\frac{-1}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \frac{-1}{2}\end{align*}
- .67 ____ -.67
Take a few minutes to check your work.
Real Life Example Completed
The Pen pal Project
The students in Mrs. Harris’ Language Arts class in New York City are participating in a pen pal project. Last summer, Mrs. Harris went to Auckland, New Zealand. While there, she participated in a conference with other educators and became good friends with another teacher from New Zealand. Mrs. Harris has arranged for her students to exchange letters with other students from New Zealand. They will be able to do some of their correspondence on email, and some will be done the old fashioned way. The students are very excited. Once a week, each class will be given some topic to investigate so that they can learn about the similarities and differences between life in New Zealand and life in the United States.
The first thing that students are being asked to look at is temperature. New Zealand and New York are in different parts of the world. The students need to figure out the average high and low winter temperatures in New Zealand and compare them with the average high and low winter temperatures in New York.
The first thing that the students notice is that the winter is New Zealand is during the months of June, July and August - the opposite of New York. The students find this very funny and Mrs. Harris takes the opportunity to teach the students about different hemispheres.
The average winter temperature in New York is:
High \begin{align*}= 40^\circ \ F\end{align*}
Low \begin{align*}= 28^\circ \ F\end{align*}
The average winter temperature is New Zealand is:
High \begin{align*}= 59^\circ \ F\end{align*}
Low \begin{align*}= 48^\circ \ F\end{align*}
One of the students, Karen, does this calculation.
The difference between the winter high and low temperatures in New York and the winter high and low temperatures in New Zealand is
\begin{align*}40 - 59 &= -19^\circ\\ 28 - 48 &= -20^\circ\end{align*}
Joey looked at her calculation and was puzzled.
“What does that mean?” he asked.
“Those are negative numbers,” Karen explained.
“Negative what?”
“Negative numbers,” Karen began.
Now that you have learned about positive and negative integers, let’s hear Karen’s explanation.
“Losses and gains can be shown in positive and negative numbers. If we were showing a gain in temperature, our answer would have been in a positive number. Because we are showing a loss in temperature by looking at the difference between New York and New Zealand, our temperature difference is a negative number. The high temperature in New York during the winter is 19 degrees less than New Zealand.”
This is just the beginning. Throughout this chapter, you will learn all about integers. A basic understanding is necessary to get started.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Integers
- the set of whole numbers and their opposites
- Negative Numbers
- numbers that are less than zero
- Positive Numbers
- numbers that are greater than zero
- Zero
- is a part of the set of integers, but is neither positive or negative
Technology Integration
Khan Academy, Locate Integers on a Number Line
James Sousa, Introduction to Integers
James Sousa, Example of Determining the Opposite of Integers
James Sousa, Example of Ordering Integers from Least to Greatest
James Sousa, Integer Application: Feet Below Sea Level
Time to Practice
Directions: Write an integer to represent each situation.
1. A loss of 20 points
2. A gain of 14 points
3. A profit of $20.00
4. A loss of $18.00
5. An elevation of 500 ft.
6. 200 feet below sea level
7. 8 degrees below zero
8. 78 degrees
9. A decrease of $68.00
10. An increase of $55.00
Directions: Write the opposite of each integer described or written
11. A loss of 18
12. A gain of 22
13. -78
14. 999
15. -87
16. 30 feet below the surface of the ocean
Directions: Compare each pair of integers using the symbols for greater than and less than.
17. 18 ____ 22
18. -12 ____ 12
19. -14 ____ -16
20. -20 ____ -33
21. 19 ____ -1
22. 0 ____ -3
23. -27 ____ -28
24. -233 ____ -300
25. -765 ____ -745
Directions: Write the following integers in order from least to greatest.
26. -4, -12, -19, -8, 0, -2, -1
27. 5, 7, 23, 8, -9, -11
28. \begin{align*}\frac{-1}{2}, \frac{-1}{4}, \frac{-5}{6}, \frac{-3}{4}\end{align*}