Have you ever wondered about time zones?

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 Concept and Karen will finish explaining at the end of the Concept.**

### Guidance

In mathematics so far, you have learned about some different kinds of numbers. You learned about whole numbers, fractions, decimals and percentages. In this Concept, 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 think about temperature.**

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

**should be written using the - sign.**

*negative number*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.**

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.**

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.**

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.

#### Example A

50 feet below sea level

**Solution: \begin{align*}-50\end{align*}**

#### Example B

$100.00 was spent

**Solution: \begin{align*}-100\end{align*}**

#### Example C

A heat wave of \begin{align*}98^\circ\end{align*}

**Solution: \begin{align*}+98\end{align*}**

Remember the pen pals?

**Now that you have learned about positive and negative integers, let’s hear Karen’s explanation.**

“What is a negative number?” Joey asked.

“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.”

### Vocabulary

Here are the vocabulary words that are found in this Concept.

- 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

### Guided Practice

Here is one for you to try on your own.

Which one of the following represents a negative integer?

No gain or loss in the stock market

The average plummeted 15 points

The overall rise was 30%

**Answer**

To figure this out, you have to look at each situation.

The first one indicates no loss or gain. This is the same as zero.

The second one indicates "plummeted" that is a negative word indicating a loss.

**This is our answer.**

### Video Review

Here is a video for review.

James Sousa, Introduction to Integers

### 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

11. The dow did not rise or fall today.

12. The plane descended at a rate of 15%.

13. The submarine surfaced.

14. The cost and the earnings evened out.

15. The stock market had an overall rate of 18% above.