# 4.1: Integers that Represent Different Situations

**At Grade**Created by: CK-12

**Practice**Integers that Represent Different Situations

Have you ever been to Alaska? Well, this state has everything, mountains, rivers, streams, wild animals, vast wilderness, many animals and extreme temperatures.

Cameron and his family are going to Alaska on vacation. To prepare for the trip, Cameron has been doing some research on Alaska. In his research, he discovered that the lowest temperature ever recorded in Alaska was in a place called Tanana Alaska and the temperature reached 78 degrees below zero one day. The highest temperature ever recorded was in 1915 when it reached over 100 degrees.

Cameron wants to write these statistics in a simple way. Do you know how he can do this?

Cameron can use integers to express these real - world situations.

**In this Concept, you will learn how to write integers to represent real - world situations. Then you will be able to help Cameron with his work.**

### Guidance

**Numbers can be classified in many different ways. 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.
- Profits and losses of money
- Locations above and below sea level.

**First, let's take a look at how integers can help us represent temperatures.**

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. We use this measurement all the time. Think of the winter or the summer. **When it is very cold or very hot, integers help us to understand how cold or how hot it is.**

**Here is one about fishing.**

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!!**

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 situations and write down any key words that you notice.**

Write an integer for each example.

#### Example A

An increase of $200.00

**Solution: \begin{align*}+200.00\end{align*}**

#### Example B

Down 10%

**Solution:\begin{align*}-10%\end{align*}**

#### Example C

50 feet below sea level

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

Now back to Cameron and his Alaska research. Here is the original problem once again.

Have you ever been to Alaska? Well, this state has everything, mountains, rivers, streams, wild animals, vast wilderness, many animals and extreme temperatures.

Cameron and his family are going to Alaska on vacation. To prepare for the trip, Cameron has been doing some research on Alaska. In his research, he discovered that the lowest temperature ever recorded in Alaska was in a place called Tanana Alaska and the temperature reached 78 degrees below zero one day. The highest temperature ever recorded was in 1915 when it reached over 100 degrees.

Cameron wants to write these statistics in a simple way. Do you know how he can do this?

Cameron can use integers to express these real - world situations.

To express these two scenarios as integers, we can write each using positive and negative signs.

The lowest temperature can be written as \begin{align*}-78^\circ\end{align*}.

The highest temperature can be written as \begin{align*}+100^\circ\end{align*}.

**This is a way to express real - world situations using integers.**

### Vocabulary

Here are the vocabulary words used in this Concept.

- 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

### Guided Practice

Here is one for you to try on your own.

Express the following situation using integers.

John gained fifteen dollars, but then he lost 20 dollars.

**Answer**

To begin, we use a positive integer to represent the gain and a negative integer to represent the loss.

\begin{align*}+15\end{align*} represents the gain.

\begin{align*}-20\end{align*} represents the loss.

**This is our answer.**

### Video Review

Here is a video for review.

- This James Sousa video is an introduction to integers.

### 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: Use two integers and an operation to represent each situation.

7. A loss of 20 and a gain of 15.

8. A gain of 15, then a loss of twenty.

9. A gain of 20 and a gain of 35.

10. A loss of 19 and another loss of 30.

11. An altitude of 30,000 feet and another height of 15,000 feet

12. A loss of 15,000 feet and another loss of 12,000 feet.

13. A depth of 60 feet and a resting stop at 15 feet below the surface of the water.

14. A depth of 15 feet and then another descent of 20 feet below the surface of the water.

15. The hikers climbed to 1500 feet and then hiked another 1000 feet.

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).Difference

The result of a subtraction operation is called a difference.fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a*rational number*.

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...### Image Attributions

Here you'll learn to write integers representing real-world situations.

## Concept Nodes:

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).Difference

The result of a subtraction operation is called a difference.fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a*rational number*.

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...