### Key Concepts

- Combine opposite quantities to make 0
- Combine opposite quantities
- Represent change using integers

### Prior Knowledge:

The elevation of a location describes its height above or below the sea level, which has an elevation of 0. Elevations below the sea level are represented by negative numbers, and elevations above the sea level are represented by the positive numbers.

- The table shows the elevations of several locations in a state park.

Graph the locations on the number line according to their elevations.

- What point on the number line represents sea level?

- Which location is closest to sea level?

- Which two locations are the same distance from sea level? Are these locations above or below sea level?

- Which location has the least elevation?

**Answers:**

- Locating the given locations on the number line below.

- ‘0’ is the point on the number line which represents sea level.

- Juniper Trail is closest to sea level because it is 3 ft below sea level.

- Little Butte and Cradle Creek are the two locations that are the same distance from sea level. Little Butte is 5 ft above sea level, and Cradle Creek is 5 ft below sea level.

- Dinosaur Valley has the least elevation.

**1.1.1 Combine Opposite Quantities to Make 0**

#### Positive Numbers:

Positive numbers are numbers that are greater than 0. Positive numbers can be written with or without a plus sign (+).

For instance, 5 is the same as +5.

#### Negative Numbers:

Negative numbers are numbers that are less than 0. Negative numbers are always written with a negative sign (-).

For instance, an integer that is 3 less than 0 is -3.

#### Opposites:

Two numbers are said to be opposite if, on a number line, they are the same distance from 0 but on different sides of 0.

For instance, 5 and -5 are opposites. 0 is its own opposite.

Integers are the set of all whole numbers and their opposites.

**Real-World Example:**

Positive and negative numbers can be used to represent real–world quantities.

For instance, 3 can represent a temperature that is 3 ^{o}F above 0. – 3 can represent a temperature that is 3 ^{o}F below 0.

Here from the above picture,

3 represents the positive temperature

– 3 represents the negative temperature

Both the quantities are opposite quantities that combine to make 0.

I.e., -3 + 3 = 0

**1.1.2 Combine Opposite Quantities**

**Examples:**

-5 and5 are opposite numbers.

The opposite of 0 is 0

Daniel kept track of the weekly low temperature in his town for several weeks. The table shows the low temperature in °F for each week.

**Example 1:**

Graph the temperature from Week 3 and it’s opposite on a number line. What do the numbers represent?

**Step 1:**

The value from Week 3 is -4.

Graph a point 4 units below 0.

**Step 2:**

Graph the opposite of -4.

Graph a point 4 units above 0.

The opposite of -4 is 4.

-4 represents a temperature that is 4 °F below 0, and 4 represents a temperature that is 4 °F above 0.

**Example 2:**

The value for Week 5 is the opposite of the opposite of the value from Week 1. What was the low temperature in Week 5?

**Step 1:**

Graph the value from Week 1 on the number line.

The value from Week 1 is -1.

**Step 2:**

Graph the opposite of -1.

The opposite of -1 is 1.

**Step 3:**

Graph the opposite of 1.

The opposite of 1 is -1.

The opposite of the opposite of -1 is -1. The low temperature in Week 5 was -1 °F.

**1.1.3 Represent Change Using Integers**

One winter morning, the temperature was –2 ^{o}C. By 11:00 AM, the temperature had decreased by 3 ^{o}C. At 4:00 PM the temperature reached 0 ^{o}C. What integer represents the temperature change from 11:00 AM to 4:00 PM?

Start at –2; the integer –3 represents the temperature decrease, so move 3 units left. The temperature has a change of –3.

Move 5 units right to show the temperature increase to 0 ^{o}C. The temperature has a change of 5.

At 11:00 AM, the temperature was –5 ^{o}C, and at 4:00 PM, the temperature was 0 ^{o}C.

∴The integer 5 represents the temperature from 11:00 AM to 4:00 PM.

## Exercise:

- Locate the number 8 and its opposite on the number line. Explain how they are related to zero.
- On a number line, locate and label 40 ℃ below zero and 40 ℃ above zero. What does zero represent in this situation?
- How far is −7 from 0, and in which direction?
- Death Valley, California, has the lowest elevation in the United States. Its elevation is 282 feet below sea level. Mount McKinley, Alaska, has the highest elevation in the United States. Its elevation is 20,320 feet above sea level. Use integers to describe these two locations in the United States.

**Chemistry:** Atoms normally have an electric charge of 0. Certain conditions, such as static, can cause atoms to have a positive or a negative charge. Atoms with a positive or negative charge are called ions

Ion | A | B | C | D | E |

Charge | -3 | +1 | -2 | +3 | -1 |

- Which ions have a negative charge?
- Which ions have charges that are opposites?
- Which ion’s charge is not the opposite of another ion’s charge?
- Which number is farther from 0 on a number line: -9 or 6? Explain your reasoning.
- Explain how to use a number line to find the opposites of the integers 3 units away from -7.
- Several wrestlers are trying to lose weight for a competition. Their change in weight since last week is shown in the chart.

- Did Victor lose or gain weight since last week?
- Which wrestler’s weight change is the opposite of Ramsey’s?
- Which wrestlers have lost weight since last week?
- Frankie’s weight change since last week was the opposite of Victor’s.
- What was Frankie’s weight change?
- Frankie’s goal last week was to gain weight. Did he meet his goal? Explain.

### What we have learned:

- Students use positive and negative numbers to indicate a change (gain or loss) in elevation with a fixed reference point, temperature, and the balance in a bank account.
- Students choose an appropriate scale for the number line when given a set of positive and negative numbers to the graph.
- Students understand that each nonzero integer,
**a**, has an opposite, denoted**-a**; and that**a**and**-a**are opposites if they are on opposite sides of zero and are the same distance from zero on the number line. - Students recognize that the number zero is its own opposite.
- Students understand that since all counting numbers are positive, it is not necessary to indicate such with a plus sign.

### Concept Map

POSITIVE NUMBERS

Numbers greater than zero. On a number line they are to the right of 0

NEGATIVE NUMBERS

Numbers less than zero. On a number line they are to the left of 0

OPPOSITE

Two integers that are each the same distance away from zero, but on opposite sides of the number line.

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