2.1 Plotting Points on the Number Line

Narrated lesson: Plotting Points on the Number Line

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Locate Fractions on the Number Line

Let’s locate fractions on the number line.

We’ll locate $\frac{1}{5}, - \frac{4}{5}, 3, \frac{7}{4}, - \frac{9}{2}, −5, and \frac{8}{3}$ on the number line.

We’ll start with the integers $3$ and $−5.$ because they are the easiest to plot. See Figure 2.1.

The proper fractions listed are $\frac{1}{5} and - \frac{4}{5} .$ We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $0 and 1.$ The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} .$ We plot $\frac{1}{5} .$ See Figure 2.1.

Similarly, $- \frac{4}{5}$ is between 0 and $−1.$ After dividing the unit into 5 equal parts we plot $- \frac{4}{5} .$ See Figure 2.1.

Finally, look at the improper fractions $\frac{7}{4}, - \frac{9}{2}, \frac{8}{3} .$ These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure 2.1.

$\begin{matrix} \frac{7}{4} = 1 \frac{3}{4} & - \frac{9}{2} = −4 \frac{1}{2} & \frac{8}{3} = 2 \frac{2}{3} \end{matrix}$

Figure 2.1 shows the number line with all the points plotted.

There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 5, negative 9/2, negative 4/5, 1/5, 4/5, 7/4, 8/3, and 3. The number negative 9/2 is halfway between negative 5 and negative 4. The number negative 4/5 is slightly to the right of negative 1. The number 1/5 is slightly to the right of 0. The number 4/5 is slightly to the left of 1. The number 7/4 is between 1 and 2, closer to 2. The number 8/3 is between 2 and 3, but a little closer to 3. — **Figure 2.1**

In Example 2, we’ll use the inequality symbols to order fractions.

a < b “a is less than b” when a is to the left of b on the number line
a > b “a is greater than b” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Example 2

Order each of the following pairs of numbers, using < or >. It may be helpful to refer to Figure 2.2.

ⓐ $- \frac{2}{3}___−1$ ⓑ $−3 \frac{1}{2}___−3$ ⓒ $- \frac{3}{4}___- \frac{1}{4}$ ⓓ $−2___- \frac{8}{3}$

Figure 2.2

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**Table 2.1**
ⓐ $- \frac{2}{3}$ is to the right of $−1$ on the number line.	$\begin{matrix} - \frac{2}{3}___−1 \\ - \frac{2}{3} > −1 \end{matrix}$
ⓑ $−3 \frac{1}{2}$ is to the left of $−3$ on the number line.	$\begin{matrix} −3 \frac{1}{2}___−3 \\ −3 \frac{1}{2} < −3 \end{matrix}$
ⓒ $- \frac{3}{4}$ is to the left of $- \frac{1}{4}$ on the number line.	$\begin{matrix} - \frac{3}{4}___- \frac{1}{4} \\ - \frac{3}{4} < - \frac{1}{4} \end{matrix}$
ⓓ $−2$ is to the right of $- \frac{8}{3}$ on the number line.	$\begin{matrix} −2___- \frac{8}{3} \\ −2 > - \frac{8}{3} \end{matrix}$

Practice Makes Perfect

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

$\frac{3}{4}, \frac{8}{5}, \frac{10}{3}$

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There is a number line shown that runs from 0 to 6. From left to right the points read 3/4, 8/5, and 10/3. The point for 3/4 is between 0 and 1. The point for 8/5 is between 1 and 2. The point for 10/3 is between 3 and 4.

$\frac{1}{4}, \frac{9}{5}, \frac{11}{3}$

$\frac{3}{10}, \frac{7}{2}, \frac{11}{6}, 4$

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There is a number line shown that runs from 0 to 6. From left to right the points read 3/10, 11/6, 7/2, and 4. The point for 3/10 is between 0 and 1. The point for 11/6 is between 1 and 2. The point for 7/2 is between 3 and 4.

$\frac{7}{10}, \frac{5}{2}, \frac{13}{8}, 3$

$\frac{2}{5}, - \frac{2}{5}$

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There is a number line shown that runs from negative 1 to 1. From left to right the points read negative 2/5 and 2/5. The point for negative 2/5 is between negative 1 and 0. The point for 2/5 is between 0 and 1.

$\frac{3}{4}, - \frac{3}{4}$

$\frac{3}{4}, - \frac{3}{4}, 1 \frac{2}{3}, −1 \frac{2}{3}, \frac{5}{2}, - \frac{5}{2}$

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There is a number line shown that runs from negative 4 to 4. From left to right the points read negative 5/2, negative 1 and 2/3, negative 3/4, ¾, 1 and 2/3, and 5/2. The point for negative 5/2 is between negative 3 and negative 2. The point for negative 1 and 2/3 is between negative 2 and negative 1. The point for negative 3/4 is between negative 1 and 0. The point for 3/4 is between 0 and 1. The point for 1 and 2/3 is between 1 and 2. The point for 5/2 is between 2 and 3.

$\frac{2}{5}, - \frac{2}{5}, 1 \frac{3}{4}, −1 \frac{3}{4}, \frac{8}{3}, - \frac{8}{3}$

In the following exercises, order each of the pairs of numbers, using < or >.

$−1___- \frac{1}{4}$

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$−1___- \frac{1}{3}$

$−2 \frac{1}{2}___−3$

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$−1 \frac{3}{4}___−2$

$- \frac{5}{12}___- \frac{7}{12}$

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$- \frac{9}{10}___- \frac{3}{10}$

$−3___- \frac{13}{5}$

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$−4___- \frac{23}{6}$

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