Plotting rational numbers on number lines in a math worksheet.

To solve the problem of plotting the given rational numbers on the number line, we need to carefully place each number according to its value. Let's go through each question step by step.

---

1. Plot $-\frac{1}{3}$ and $1 \frac{2}{3}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|---|--->
-1 -0.5 0 0.5 1 1.5 2
```

#### Steps:
1. $-\frac{1}{3}$:
- This is a negative number, so it lies to the left of 0.
- $\frac{1}{3} \approx 0.33$, so $-\frac{1}{3} \approx -0.33$.
- Place it one-third of the way from 0 to -1.

2. $1 \frac{2}{3}$:
- Convert $1 \frac{2}{3}$ to an improper fraction: $1 \frac{2}{3} = \frac{5}{3}$.
- $\frac{5}{3} \approx 1.67$.
- Place it two-thirds of the way from 1 to 2.

#### Final Plot:
```
<---|---|---|---|---|---|---|--->
-1 -0.5 0 0.5 1 1.5 2
| |
-1/3 1 2/3
```

---

2. Plot $\frac{2}{3}$ and $1 \frac{1}{3}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|--->
0 0.5 1 1.5 2
```

#### Steps:
1. $\frac{2}{3}$:
- $\frac{2}{3} \approx 0.67$.
- Place it two-thirds of the way from 0 to 1.

2. $1 \frac{1}{3}$:
- Convert $1 \frac{1}{3}$ to an improper fraction: $1 \frac{1}{3} = \frac{4}{3}$.
- $\frac{4}{3} \approx 1.33$.
- Place it one-third of the way from 1 to 2.

#### Final Plot:
```
<---|---|---|---|---|---|--->
0 0.5 1 1.5 2
| |
2/3 1 1/3
```

---

3. Plot $-\frac{1}{2}$ and $1 \frac{3}{4}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|--->
-1 -0.5 0 0.5 1 1.5 2
```

#### Steps:
1. $-\frac{1}{2}$:
- This is a negative number, so it lies to the left of 0.
- $-\frac{1}{2} = -0.5$.
- Place it exactly halfway between 0 and -1.

2. $1 \frac{3}{4}$:
- Convert $1 \frac{3}{4}$ to an improper fraction: $1 \frac{3}{4} = \frac{7}{4}$.
- $\frac{7}{4} = 1.75$.
- Place it three-fourths of the way from 1 to 2.

#### Final Plot:
```
<---|---|---|---|---|---|--->
-1 -0.5 0 0.5 1 1.5 2
| |
-1/2 1 3/4
```

---

4. Plot $\frac{1}{4}$ and $1 \frac{2}{4}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|--->
0 0.5 1 1.5 2
```

#### Steps:
1. $\frac{1}{4}$:
- $\frac{1}{4} = 0.25$.
- Place it one-fourth of the way from 0 to 0.5.

2. $1 \frac{2}{4}$:
- Simplify $1 \frac{2}{4}$: $1 \frac{2}{4} = 1 \frac{1}{2} = \frac{3}{2}$.
- $\frac{3}{2} = 1.5$.
- Place it exactly at 1.5.

#### Final Plot:
```
<---|---|---|---|---|---|--->
0 0.5 1 1.5 2
| |
1/4 1 2/4
```

---

5. Plot $-1 \frac{1}{2}$ and $1 \frac{4}{6}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|---|--->
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
```

#### Steps:
1. $-1 \frac{1}{2}$:
- Convert $-1 \frac{1}{2}$ to an improper fraction: $-1 \frac{1}{2} = -\frac{3}{2}$.
- $-\frac{3}{2} = -1.5$.
- Place it exactly at -1.5.

2. $1 \frac{4}{6}$:
- Simplify $1 \frac{4}{6}$: $1 \frac{4}{6} = 1 \frac{2}{3} = \frac{5}{3}$.
- $\frac{5}{3} \approx 1.67$.
- Place it two-thirds of the way from 1 to 2.

#### Final Plot:
```
<---|---|---|---|---|---|---|--->
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
| |
-1 1/2 1 4/6
```

---

6. Plot $1 \frac{1}{3}$ and $2 \frac{5}{6}$ on the number line below

#### Number Line:
```
<---|---|---|---|---|---|---|---|---|---|--->
0 0.5 1 1.5 2 2.5 3
```

#### Steps:
1. $1 \frac{1}{3}$:
- Convert $1 \frac{1}{3}$ to an improper fraction: $1 \frac{1}{3} = \frac{4}{3}$.
- $\frac{4}{3} \approx 1.33$.
- Place it one-third of the way from 1 to 2.

2. $2 \frac{5}{6}$:
- Convert $2 \frac{5}{6}$ to an improper fraction: $2 \frac{5}{6} = \frac{17}{6}$.
- $\frac{17}{6} \approx 2.83$.
- Place it five-sixths of the way from 2 to 3.

#### Final Plot:
```
<---|---|---|---|---|---|---|---|---|---|--->
0 0.5 1 1.5 2 2.5 3
| |
1 1/3 2 5/6
```

---

Final Answer:

$$
\boxed{
\text{Plots as described above.}
}
$$