Lesson 1: Positive and Negative Numbers

Let’s explore how we represent temperatures and elevations.

1.1: Notice and Wonder: Memphis and Bangor

What do you notice? What do you wonder?

1.2: Above and Below Zero

The thermometer showed a temperature of 7 degrees Celsius one morning. Later, the temperature increased 4 degrees.

We can represent this change in temperature using a number line, as shown here.

  1. Here are three situations involving changes in temperature. Represent each change on the applet, and draw it on a number line. Then, answer the question.

    1. At noon, the temperature was 5 degrees Celsius. By late afternoon, it has risen 6 degrees Celsius. What was the temperature late in the afternoon?

    2. The temperature was 8 degrees Celsius at midnight. By dawn, it has dropped 12 degrees Celsius. What was the temperature at dawn?

    3. Water freezes at 0 degrees Celsius, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7 degrees Celsius. What is the freezing temperature of the gallon of salt water?

GeoGebra Applet csdsSV2n

  1. Discuss with a partner:
    1. How did you name the resulting temperature in each situation? Did both of you refer to each resulting temperature by the same name or different names?
    2. What does it mean when the resulting temperature is above 0 on the number line? What does it mean when a temperature is below 0?
    3. Do numbers below 0 make sense outside of the context of temperature? If you think so, give some examples to show how they make sense. If you don’t think so, give some examples to show otherwise.

1.3: High Places, Low Places

  1. Here is a table that shows elevations of various cities.

      city elevation (feet)
    row 1 Harrisburg, PA 320
    row 2 Bethell, IN 1,211
    row 3 Denver, CO 5,280
    row 4 Coachella, CA -22
    row 5 Death Valley, CA -282
    row 6 New York City, NY 33
    row 7 Miami, FL 0
    1. On the list of cities, which city has the second highest elevation?
    2. How would you describe the elevation of Coachella, CA, in relation to sea level?
    3. How would you describe the elevation of Death Valley, CA, in relation to sea level?
    1. If you are standing on a beach right next to the ocean, what is your elevation?
       
    2. How would you describe the elevation of Miami, FL?

    1. A city has a higher elevation than Coachella, CA. Select all numbers that could represent the city’s elevation. Be prepared to explain your reasoning.

      -11 feet

      -35 feet

      4 feet

      -8 feet

      0 feet

       

  2. Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level. Drag the points marking the mountains and trenches to the vertical number line and answer the questions.

      point mountain continent elevation (meters)
    row 1 C Everest Asia 8,848
    row 2 H Kilimanjaro Africa 5,895
    row 3 E Denali North America 6,168
    row 4 A Pikchu Pikchu South America 5,664
      point trench ocean elevation (meters)
    row 1 F Mariana Trench Pacific -11,033
    row 2 B Puerto Rico Trench Atlantic -8,600
    row 3 D Tonga Trench Pacific -10,882
    row 4 G Sunda Trench Indian -7,725

    GeoGebra Applet cp3xmE5v

    1. Which point in the ocean is the lowest in the world? What is its elevation?
    1. Which mountain is the highest in the world? What is its elevation?
    1. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0?
    2. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.

Summary

Positive numbers are numbers that are greater than 0. Negative numbers are numbers that are less than zero. The meaning of a negative number in a context depends on the meaning of zero in that context.

For example, if we measure temperatures in degrees Celsius, then 0 degrees Celsius corresponds to the temperature at which water freezes.

In this context, positive temperatures are warmer than the freezing point and negative temperatures are colder than the freezing point. A temperature of -6 degrees Celsius means that it is 6 degrees away from 0 and it is less than 0. This thermometer shows a temperature of -6 degrees Celsius.

If the temperature rises a few degrees and gets very close to 0 degrees without reaching it, the temperature is still a negative number. 

A vertical thermometer with the numbers negative 8 through 8 indicated. The thermometer is shaded starting from the bottom of the thermometer to negative 6.

 

Another example is elevation, which is a distance above or below sea level. An elevation of 0 refers to the sea level. Positive elevations are higher than sea level, and negative elevations are lower than sea level.

Practice Problems ▶

Glossary

negative number

negative number

A negative number is any number that is less than zero. On a number line, negative numbers are always on the opposite side of 0 as 1. If a number line is drawn horizontally, the negative numbers are usually to the left of zero.

If a number line is drawn vertically, then the negative numbers are usually below zero, like most thermometers or on the vertical axis in the coordinate plane.

positive number

positive number

A positive number is any number that is greater than zero. On a number line, positive numbers are always on the same side of 0 as 1. If a number line is drawn horizontally, the positive numbers are usually to the right of zero.

If a number line is drawn vertically, then the positive numbers are usually above zero, like most thermometers or on the vertical axis in the coordinate plane.