Lesson 2: Points on the Number Line

Let’s plot positive and negative numbers on the number line.

2.1: A Point on the Number Line

Which of the following numbers could be $B$?

2.5

$\frac25$

$\frac{5}{2}$

$\frac{25}{10}$

2.49

 

2.2: 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 are labeled a, b, c, d, and e.  Thermometer a has the numbers negative 4 to 5 indicated. The number 4 is on the bottom and 5 is on the top. The thermometer is shaded from the bottom of the thermometer to 1. Thermometer b has the numbers negative 4 to 5 indictaed. The number 4 is on the bottom and 5 is on the top. The thermometer is shaded from the bottom of the thermometer to negative 2.  Thermometer c has the numbers negative 4 to 5 indicated. The number 4 is on the bottom and 5 is on the top. The thermometer is shaded from the bottom of the thermometer to halfway between 3 and 4.  Thermometer d has the numbers negative 4 to 5 indicated. The number 4 is on the bottom and 5 is on the top. The thermometer is shaded from the bottom of the thermometer to halfway between negative 1 and zero.  Thermometer e has the numbers negative 20, negative 5, 10, and 25 indicated. There are 2 evenly spaced tick marks with missing labels between negative 20 and negative 5, between negative 5 and 10, and between 10 and 25. The thermometer is shaded from the bottom of the thermometer to negative 20.
    The last thermometer is missing some numbers. Write them in the boxes.

  1. 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.
  2. 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?

A thermometer is labeled from negative 4 through 5 with negative at the bottom of the thermometer. There are tick marks halfway between each integer. The thermometer is shaded from the bottom of the thermometer to the tick mark between negative 2 and negative 1.

2.3: Folded Number Lines

Your teacher will give you a sheet of tracing paper on which to draw a number line. 

  1. Follow the steps to make your own number line.

    • Use a straightedge or a ruler to draw a horizontal line. Mark the middle point of the line and label it 0.
    • To the right of 0, draw tick marks that are 1 centimeter apart. Label the tick marks 1, 2, 3. . . 10. This represents the positive side of your number line.
    • Fold your paper so that a vertical crease goes through 0 and the two sides of the number line match up perfectly.
    • Use the fold to help you trace the tick marks that you already drew onto the opposite side of the number line. Unfold and label the tick marks -1, -2, -3. . . -10. This represents the negative side of your number line.
  2. Use your number line to answer these questions:

    1. Which number is the same distance away from zero as is the number 4?
    1. Which number is the same distance away from zero as is the number -7?
    1. Two numbers that are the same distance from zero on the number line are called opposites. Find another pair of opposites on the number line.
    2. Determine how far away the number 5 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number 5.
    3. Determine how far away the number -2 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number -2.

    Pause here so your teacher can review your work.

  3. Here is a number line with some points labeled with letters. Determine the location of points $P$, $X$, and $Y$.

    Three points labeled P, X, and Y on a number line. The number 0 is located on the middle of the number line. To the right of 0, the numbers 1 through 10 are indicated on 10 evenly spaced tick marks. To the left of 0, points P, X and Y are plotted. When the number line is folded at 0, point P aligns with 2, point X is halfway between 4 and 5, and point Y is halfway between 7 and 8.

    If you get stuck, trace the number line and points onto a sheet of tracing paper, fold it so that a vertical crease goes through 0, and use the folded number line to help you find the unknown values.

Summary

Here is a number line labeled with positive and negative numbers. The number 4 is positive, so its location is 4 units to the right of 0 on the number line. The number -1.1 is negative, so its location is 1.1 units to the left of 0 on the number line.

A number line with 11 evenly spaced tick marks, the numbers negative 5 through 5 indicated.

We say that the opposite of 8.3 is -8.3, and that the opposite of $\frac {\text{-}3}{2}$ is $\frac32$. Any pair of numbers that are equally far from 0 are called opposites.

Points $A$ and $B$ are opposites because they are both 2.5 units away from 0, even though $A$ is to the left of 0 and $B$ is to the right of 0.

Three points A, B, and C plotted on a number line with the numbers negative 5 through 5 indicated.  Point A is halfway between negative 3 and negative 2. Point B is halfway between 2 and 3. Point C is halfway between 4 and 5.

A positive number has a negative number for its opposite. A negative number has a positive number for its opposite. The opposite of 0 is itself.

You have worked with positive numbers for many years. All of the positive numbers you have seen—whole and non-whole numbers—can be thought of as fractions and can be located on a the number line.

To locate a non-whole number on a number line, we can divide the distance between two whole numbers into fractional parts and then count the number of parts. For example, 2.7 can be written as $2 \frac {7}{10}$. The segment between 2 and 3 can be partitioned into 10 equal parts or 10 tenths. From 2, we can count 7 of the tenths to locate 2.7 on the number line.

All of the fractions and their opposites are what we call rational numbers. For example, 4, -1.1, 8.3, -8.3, $\frac {\text{-}3}{2}$, and $\frac32$ are all rational numbers.

Practice Problems ▶

Glossary

opposite

opposite

Two numbers are opposites of each other if they are the same distance from 0 on the number line, but on opposite sides.

rational number

rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into $b$ equal parts and finding the point that is $a$ of them from 0. We can always write a fraction in the form $\frac{a}{b}$ where $a$ and $b$ are whole numbers, with $b$ not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as $\frac{7}{10}$.