Lesson 1

Positive and Negative Numbers

Let’s explore how we represent temperatures and elevations.

1.1: Notice and Wonder: Memphis and Bangor

Two weather reports, Memphis, Saturday 5 p m, light rain showers, 37 degrees Farenheit, 3 degrees Celcius. Bangor, Saturday 6 p m, partly cloudy, 1 degree Farenheit, negative 17 degrees celsius.

What do you notice? What do you wonder?

 

1.2: What’s the Temperature?

  1. Here are five thermometers. The first four thermometers show temperatures in Celsius. Write the temperatures in the blanks.

    Five vertical thermometers, a through e.

    The last thermometer is missing some numbers. Write them in the boxes.

  2. Elena says that the thermometer shown here reads \(\text-2.5^\circ \text{C}\) because the line of the liquid is above \(\text-2^\circ \text{C}\). Jada says that it is \(\text-1.5^\circ \text{C}\). Do you agree with either one of them? Explain your reasoning.

    A thermometer labeled negative 4 to 5 by ones. It is shaded up to the halfway point between negative 2 and negative 1.
  3. One morning, the temperature in Phoenix, Arizona, was \(8^\circ \text{C}\) and the temperature in Portland, Maine, was \(12^\circ \text{C}\) cooler. What was the temperature in Portland?

1.3: Seagulls Soar, Sharks Swim

Here is a picture of some sea animals. The number line on the left shows the vertical position of each animal above or below sea level, in meters.

A graph. Vertical position, meters. 

  1. How far above or below sea level is each animal? Measure to their eye level.

  2. A mobula ray is 3 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:

    The jumping dolphin?

    The flying seagull?

    The octopus?

  3. An albatross is 5 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:

    The jumping dolphin?

    The flying seagull?

    The octopus?

  4. A clownfish is 2 meters below the surface of the ocean. How does its vertical position compare to the height or depth of:

    The jumping dolphin?

    The flying seagull?

    The octopus?

  5. The vertical distance of a new dolphin from the dolphin in the picture is 3 meters. What is its distance from the surface of the ocean?



The north pole is in the middle of the ocean. A person at sea level at the north pole would be 3,949 miles from the center of Earth. The sea floor below the north pole is at an elevation of approximately -2.7 miles. The elevation of the south pole is about 1.7 miles. How far is a person standing on the south pole from a submarine at the sea floor below the north pole?

1.4: High Places, Low Places

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

    city elevation (feet)
    Harrisburg, PA 320
    Bethell, IN 1,211
    Denver, CO 5,280
    Coachella, CA -22
    Death Valley, CA -282
    New York City, NY 33
    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?
    4. If you are standing on a beach right next to the ocean, what is your elevation?
    5. How would you describe the elevation of Miami, FL?

    6. 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.

      1. -11 feet
      2. -35 feet
      3. 4 feet
      4. -8 feet
      5. 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)
    C Everest Asia 8,848
    H Kilimanjaro Africa 5,895
    E Denali North America 6,168
    A Pikchu Pikchu South America 5,664
    point trench ocean elevation (meters)
    F Mariana Trench Pacific -11,033
    B Puerto Rico Trench Atlantic -8,600
    D Tonga Trench Pacific -10,882
    G Sunda Trench Indian -7,725

    1. Which point in the ocean is the lowest in the world? What is its elevation?
    2. Which mountain is the highest in the world? What is its elevation?
    3. 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?
    4. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.


A spider spins a web in the following way:

  • It starts at sea level.
  • It moves up one inch in the first minute.
  • It moves down two inches in the second minute.
  • It moves up three inches in the third minute.
  • It moves down four inches in the fourth minute.

Assuming that the pattern continues, what will the spider’s elevation be after an hour has passed?

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.

Vertical number line, elevation, horizontal number line at zero, sea level.  Silhouettes of animal, from lowest to highest, octopus, shark, dolphin, seagull.

Glossary Entries

  • negative number

    A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

    number line with arrow that extends from 0 to -7
  • positive number

    A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

    Number line with green arrow extending from 0 to 7