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Lesson 3Comparing Positive and Negative Numbers

Let’s compare numbers on the number line.

Learning Targets:

  • I can explain how to use the positions of numbers on a number line to compare them.
  • I can explain what a rational number is.
  • I can use inequalities to compare positive and negative numbers.

3.1 Which One Doesn’t Belong: Inequalities

Which inequality doesn’t belong?

\frac{5}{4} < 2

8.5 > 0.95

8.5 < 7

10.00 < 100

3.2 Comparing Temperatures

Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.

day Mon Tues Weds Thurs Fri Sat Sun
temperature 5 -1 -5.5 -2 3 4 0
  1. Plot the temperatures on a number line. Which day of the week had the lowest low temperature?

  2. The lowest temperature ever recorded in the United States was -62 degrees Celsius, in Prospect Creek Camp, Alaska. The average temperature on Mars is about -55 degrees Celsius.

    1. Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
    2. Write an inequality to show your answer.

  3. On a winter day the low temperature in Anchorage, Alaska was -21 degrees Celsius and the low temperature in Minneapolis, Minnesota was -14 degrees Celsius.

    Jada said: “I know that 14 is less than 21, so -14 is also less than -21. This means that it was colder in Minneapolis than in Anchorage.”

    Do you agree? Explain your reasoning.

Are you ready for more?

Another temperature scale frequently used in science is the Kelvin scale. In this scale, 0 is the lowest possible temperature of anything in the universe, and it is -273.15 degrees in the Celsius scale. Each 1\text{ K} is the same as 1 ^\circ\text{C} , so 10\text{ K} is the same as \text-263.15 ^\circ\text{C} .

  1. Water boils at 100^\circ \text{C} . What is this temperature in \text{K} ?
  2. Ammonia boils at \text-35.5^\circ \text{C} . What is the boiling point of ammonia in \text{K} ?
  3. Explain why only positive numbers (and 0) are needed to record temperature in \text{K} .

3.3 Rational Numbers on a Number Line

  1. Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.
A blank number line

  1. Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.

    \text-2 < 4

    \text-2 < \text-7

    4 > \text-7

    \text-7 > 10

Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.

open applet in presentation mode

  1. Andre says that \frac14 is less than \text{-}\frac {3}{4} because, of the two numbers, \frac14 is closer to 0. Do you agree? Explain your reasoning.

  2. Answer each question. Be prepared to explain how you know.

    1. Which number is greater: \frac14 or \frac54 ?

    2. Which number is farther from 0: \frac14 or \frac54 ?

    3. Which number is greater: \text{-}\frac {3}{4} or \frac58 ?

    4. Which number is farther from 0: \text{-}\frac {3}{4} or \frac58 ?

    5. Is the number that is farther from 0 always the greater number? Explain your reasoning.

Lesson 3 Summary

We use the words greater than and less than to compare numbers on the number line. For example, the numbers -2.7, 0.8, and -1.3, are shown on the number line.

Three points plotted on a number line and the numbers negative 3 through 3 are indicated. The numbers are as follows: Point 1: negative 2 point 7. Point 2: negative 3 and negative 2. Point 3: zero point 8

Because -2.7 is to the left of -1.3, we say that -2.7 is less than -1.3. We write: \text-2.7 <\text -1.3  In general, any number that is to the left of a number n is less than n .

We can see that -1.3 is greater than -2.7 because -1.3 is to the right of -2.7. We write \text-1.3 >\text -2.7 In general, any number that is to the right of a number n is greater than n

We can also see that 0.8 > \text-1.3 and 0.8 > \text-2.7 . In general, any positive number is greater than any negative number.

Glossary Terms

sign

The sign of any number other than 0 is either positive or negative.

For example, the sign of 6 is positive. The sign of -6 is negative. Zero does not have a sign, because it is not positive or negative.

Lesson 3 Practice Problems

  1. Decide whether each inequality statement is true or false. Explain your reasoning.

    1. \text-5 > 2
    2. 3 > \text-8
    3. \text-12 > \text-15
    4. \text-12.5 > \text-12

  2. Here is a true statement: \text-8.7 < \text-8.4 . Select all of the statements that are equivalent to  \text-8.7 < \text-8.4 .

    1. -8.7 is further to the right on the number line than -8.4.

    2. -8.7 is further to the left on the number line than -8.4.

    3. -8.7 is less than -8.4.

    4. -8.7 is greater than -8.4.

    5. -8.4 is less than -8.7.

    6. -8.4 is greater than -8.7.

  3. The table shows five states and the lowest point in each state.

    state lowest elevation (feet)
    California -282
    Colorado 3350
    Louisiana -8
    New Mexico 2842
    Wyoming 3099
    Put the states in order by their lowest elevation, from least to greatest.

  4. Plot each of the following numbers on the number line. Label each point with its numeric value.

    0.4, -1.5, \text-1\frac{7}{10} , \text{-}\frac{11}{10}

    A number line with 5 evenly spaced tick marks, labeled negative 2 through 2.

  5. Each lap around the track is 400 meters.

    1. How many meters does someone run if they run:

      2 laps?

      5 laps?

      x laps?

    1. If Noah ran 14 laps, how many meters did he run?
    2. If Noah ran 7,600 meters, how many laps did he run?

  6. A stadium can seat 16,000 people at full capacity.

    1. If there are 13,920 people in the stadium, what percentage of the capacity is filled? Explain or show your reasoning.
    2. What percentage of the capacity is not filled?